On Reachability Equivalence for BPP-Nets

21 Now we show the size of the semilinear set representation as well as the time required for generating such a representation. From Condition 1, each component of is bounded by 2 d 1 s 2 ; hence jBj (i.e., the number of distinct bases of the semilinear set) is bounded by (2 d 1 s 2) k 2 d 1 s 3 (k(s) is the dimension of). As a result, the size of B is bounded by k 3 (log 2 (2 d 1 s 2)) 3 (2 d 1 s 3). Likewise, from Condition 2 the size of each period is bounded by k 3 (log 2 (2 d 2 s 2)) 3 (2 d 2 s 3). In summary, the size of S 2B L(;) is (the size of B) + X 2B (the size of), which is bounded by O(2 c 1 s 3), for some constant c 1. As for the amount of time needed to generate the semilinear set, rst recall from Theorem 4 that the reachability problem for BPP-nets in NP. From our earlier discussion, each base vector is of size k 3 (log 2 (2 d 2 s 2)), which is polynomial in O(s 3). Hence, the reachability of from 0 can be checked in DT IME(2 O(s 3)). Similarly, checking the existence of a positive loop satisfying Conditions 2 (a) and (b) can also be done in DT IME(2 O(s 3)). By exhaustive search, the desired semilinear set can be constructed in DT IME(2 c 2 s 3), for some constant c 2. 2